Filtering of nonlinear stochastic feedback systems

This paper concerns the filtering problem for a class of stochastic nonlinear systems where the drift term may depend either on some external function (open-loop system) or on the system output ( closed-loop system), through a controller. Such systems are denoted feedback systems. The following result is proven: for feedback systems, the optimal filter in the open-loop case remains optimal when the feedback is closed. The proof is obtained by showing equivalence of suitable expressions for the estimators of the open-loop and closed-loop systems, obtained using the Kallianpur--Striebel formula [G. Kallianpur and C. Striebel, Ann. Math. Statist., 39 (1968), pp. 785--801].